CBSE Class 10 Mathematics Polynomials Worksheet Set 07

SECTION - A

Multiple Choice Questions (MCQs)

Question. The zeroes of the quadratic polynomial \( x^2 + 99x + 127 \) are :
(a) both positive
(b) both negative
(c) one positive and one negative
(d) both equal
Answer: (b) both negative

 

Question. If \( \alpha, \beta \) are zeroes of \( x^2 - 4x + 1 \), then \( \frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta \) is :
(a) 3
(b) 5
(c) -5
(d) -3
Answer: (a) 3

 

Question. If \( \alpha, \beta \) are zeroes of polynomial \( f(x) = x^2 + px + q \) then polynomial having \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) as its zeroes is:
(a) \( x^2 + qx + p \)
(b) \( x^2 - px + q \)
(c) \( qx^2 + px + 1 \)
(d) \( px^2 + qx + 1 \)
Answer: (c) \( qx^2 + px + 1 \)

 

Question. If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( 5x^2 - 7x + 2 \), then sum of their reciprocals is :
(a) \( \frac{7}{2} \)
(b) \( \frac{7}{5} \)
(c) \( \frac{2}{5} \)
(d) \( \frac{14}{25} \)
Answer: (a) \( \frac{7}{2} \)

 

Question. The quadratic polynomial \( p(x) \) with -81 and 3 as product and one of the zeroes respectively is:
(a) \( x^2 + 24x - 81 \)
(b) \( x^2 - 24x - 81 \)
(c) \( x^2 - 24x + 81 \)
(d) \( x^2 + 24x + 81 \)
Answer: (a) \( x^2 + 24x - 81 \)

 

Question. If 1 is zero of the polynomial \( p(x) = ax^2 - 3(a - 1)x - 1 \), then the value of 'a' is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (a) 1

 

Question. If \( \alpha, \beta \) are zeroes of \( x^2 - 6x + k \). What is the value of \( k \) if \( 3\alpha + 2\beta = 20 \).
(a) -16
(b) 8
(c) -2
(d) -8
Answer: (a) -16

 

Question. If one zero of \( 2x^2 - 3x + k \) is reciprocal to the other, then the value of \( k \) is :
(a) 2
(b) \( \frac{-2}{3} \)
(c) \( \frac{-3}{2} \)
(d) -3
Answer: (a) 2

 

Question. The number of polynomials having zeroes -2 and 5 is :
(a) 1
(b) 2
(c) 3
(d) more than 3
Answer: (d) more than 3

 

Question. The value of \( p \) for which the polynomial \( x^3 + 4x^2 - px + 8 \) is exactly divisible by \( (x - 2) \) is
(a) 0
(b) 3
(c) 16
(d) 12
Answer: (c) 16

 

Question. The quadratic polynomial whose sum of zeroes is 3 and product of zeroes is -2 is:
(a) \( x^2 + 3x - 2 \)
(b) \( x^2 - 2x + 3 \)
(c) \( x^2 - 3x + 3 \)
(d) \( x^2 - 3x - 2 \)
Answer: (d) \( x^2 - 3x - 2 \)

 

Question. If one of the zeroes of the quadratic polynomial \( (k - 1)x^2 + kx + 1 \) is -3, then the value of \( k \) is
(a) -4/3
(b) 4/3
(c) 2/3
(d) -2/3
Answer: (b) 4/3

 

Question. The number of zeroes for the polynomial \( y = p(x) \). If graph meet x-axis at 3 points is:
(a) 3
(b) 1
(c) 2
(d) 0
Answer: (a) 3

 

Question. The degree of the polynomial \( (x + 1)(x^2 - x - x^4 + 1) \) is :
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (d) 5

 

Question. If -4 is a zero of the polynomial \( x^2 - x - (2 + 2k) \), then the value of \( k \) is
(a) 3
(b) 9
(c) 6
(d) -9
Answer: (b) 9

 

Question. The quadratic polynomial having zeroes are 1 and -2 is :
(a) \( x^2 - x + 2 \)
(b) \( x^2 - x - 2 \)
(c) \( x^2 + x - 2 \)
(d) \( x^2 + x + 2 \)
Answer: (c) \( x^2 + x - 2 \)

SECTION - B

Very Short Answer Type Questions

Question. If \( \alpha \) and \( \beta \) are the zeroes of \( x^2 + 7x + 12 \), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} + 2\alpha\beta \).
Answer: \( \frac{281}{12} \)

 

Question. Find the zeroes of the quadartic polynomial \( 2x^2 - 25 \).
Answer: \( \pm \frac{5}{\sqrt{2}} \)

 

Question. Find the zeroes of the quadratic polynomial \( 4x^2 - 7 \).
Answer: \( \pm \frac{\sqrt{7}}{2} \)

 

Question. Find a quadratic polynomial whose zeroes are \( 3 + \sqrt{5} \) and \( 3 - \sqrt{5} \).
Answer: \( x^2 - 6x + 4 \)

 

Question. If \( \alpha, \beta \) are zeroes of quadratic polynomial \( x^2 - (k + 6)x + 2(2k - 1) \). Find \( k \) if \( \alpha + \beta = \frac{1}{2}\alpha\beta \).
Answer: \( k = 7 \)

 

Question. Form a quadratic polynomial whose one of the zeroes is +15 and sum of the zeroes is 42.
Answer: \( x^2 - 42x - 405 \)

 

Question. Divide \( (2x^2 + x - 20) \) by \( (x + 3) \) and verify division algorithm.
Answer: \( Q = 2x - 5, R = -5 \)

 

Question. Find the zeroes of the quadratic polynomial \( \sqrt{3}x^2 - 8x + 4\sqrt{3} \).
Answer: \( 2\sqrt{3} \) and \( \frac{2}{\sqrt{3}} \)

 

Question. What must be added to polynomial \( f(x) = x^4 + 2x^3 - 2x^2 + x - 1 \) so that the resulting polynomial is exactly divisible by \( x^2 + 2x - 3 \)?
Answer: \( x - 2 \)

 

Question. Find a quadratic polynomial with zeroes \( 3 + \sqrt{2} \) and \( 3 - \sqrt{2} \).
Answer: \( x^2 - 6x + 7 \)

 

Question. If \( \alpha \) and \( \frac{1}{\alpha} \) are the zeroes of the polynomial \( 4x^2 - 2x + (k - 4) \), find the value of \( k \).
Answer: \( k = 8 \)

 

Question. It being given that 1 is one of the zeroes of the polynomial \( 7x - x^3 - 6 \). Find its other zeroes.
Answer: 2 and -3

 

Question. \( \alpha, \beta \) are the roots of the quadratic polynomial \( p(x) = x^2 - (k - 6)x + (2k + 1) \). Find the value of \( k \), if \( \alpha + \beta = \alpha\beta \).
Answer: \( k = -7 \)

 

Question. Find the zeroes of the polynomial \( 100x^2 - 81 \).
Answer: \( \pm \frac{9}{10} \)

 

Question. Divide the polynomial \( p(x) = 3x^2 - x^3 - 3x + 5 \) by \( g(x) = x - 1 - x^2 \) and find its quotient and remainder.
Answer: \( Q = x - 2, R = 3 \)

 

Question. Find a quadratic polynomial, the sum of whose zeroes is 7 and their product is 12. Hence find the zeroes of the polynomial.
Answer: \( x^2 - 7x + 12 \); zeroes are 3 and 4

 

Question. Find a quadratic polynomial whose zeroes are 2 and -6. Verify the relation between the coefficients and zeroes of the polynomial.
Answer: \( x^2 + 4x - 12 \)

 

Question. Can \( (x - 3) \) be the remainder on the division of a polynomial \( p(x) \) by \( (2x + 5) \)? Justify your answer.
Answer: No

 

Question. Form a quadratic polynomial \( p(y) \) with sum and product of zeroes are 2 and -3/5 respectively.
Answer: \( x^2 - 2x - \frac{3}{5} \)

 

Question. Divide \( 6x^3 + 13x^2 + x - 2 \) by \( 2x + 1 \), and find quotient and remainder.
Answer: \( Q = 3x^2 + 5x - 2, R = 0 \)

SECTION - C

Assertion-Reason Questions

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true but R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

 

Question. Assertion (A) : If \( (2 - \sqrt{3}) \) is one zero of a quadratic polynomial then other zero will be \( (2 + \sqrt{3}) \).
Reason (R) : Irrational zeros (roots) always occurs in pairs.
Answer: Solution : As irrational roots/zeros always occurs in pairs therefore, when one zero is \( 2 - \sqrt{3} \) then other will be \( 2 + \sqrt{3} \). So, both A and R are true and R is the correct explanation of A.
Hence, option (a) is correct.

 

Question. Assertion (A) : If both zeros of the quadratic polynomial \( x^2 - 2kx + 2 \) are equal in magnitude but opposite in sign then value of \( k \) is \( \frac{1}{2} \).
Reason (R) : Sum of zeros of a quadratic polynomial \( ax^2 + bx + c \) is \( \frac{-b}{a} \).
Answer: Solution : Given polynomial is \( x^2 - 2kx + 2 \) its zeros are equal but opposite in sign.
\( \therefore \text{Sum of zeros} = 0 = \frac{-(-2k)}{1} = 0 \)

\( \implies 2k = 0 \)
\( \implies k = 0 \)
So, A is false but R is true.
Hence, option (d) is correct.

 

Question. Assertion (A) : \( p(x) = 14x^3 - 2x^2 + 8x^4 + 7x - 8 \) is a polynomial of degree 3.
Reason (R) : The highest power of \( x \) in any polynomial \( p(x) \) is the degree of the polynomial.
Answer: Solution : The highest power of \( x \) in the polynomial \( p(x) = 14x^3 - 2x^2 + 8x^4 + 7x - 8 \) is 4.
\( \therefore \text{Degree of } p(x) \text{ is } 4. \text{ So, A is false but R is true.} \)
Hence, option (d) is correct.

 

Question. Assertion (A) : The graph \( y = f(x) \) is shown in figure, for the polynomial \( f(x) \). The number of zeros of \( f(x) \) is 4.
Reason (R) : The number of zero of the polynomial \( f(x) \) is the number of point at which \( f(x) \) cuts or touches the axes.
Answer: Solution : As the number of zeroes of polynomial \( f(x) \) is the number of points at which \( f(x) \) cuts (intersects) the \( x \)-axis and number of zeroes in the given figure is 4. So A is true but R is false.
Hence, option (c) is correct.

 

Question. Assertion (A) : If the sum and product of the zeros of a quadratic polynomial are \( -\frac{1}{4} \) and \( \frac{1}{4} \) respectively. Then the quadratic polynomial is \( 4x^2 + x + 1 \).
Reason (R) : The quadratic polynomial whose sum and product of zeros are given is \( x^2 - (\text{sum of zeros})x + \text{product of zeros} \).
Answer: Solution : \( \text{Sum of zeros} = -\frac{1}{4} \text{ and product of zeros} = \frac{1}{4} \)
\( \therefore \text{Quadratic polynomial is } x^2 - \left( -\frac{1}{4} \right)x + \frac{1}{4} \)

\( \implies x^2 + \frac{1}{4}x + \frac{1}{4} = \frac{1}{4}(4x^2 + x + 1) \)
\( \therefore \text{Quadratic polynomial be } 4x^2 + x + 1. \)
So, both A and R are true and R is the correct explanation of A.
Hence, option (a) is correct.